A Research On Duality And Application Of Semidefinite Programming

As an extension of linear programming in nonlinear programming,semidefinite programming is an important branch of mathematical programming.The formal proposal of semidefinite programming comes from the interior point algorithm.When we look back,we find that there have been previous studies.Semidefinite programming is widely used in Various fields,such as combination optimization,cybernetics,structural design,statistics and signal processing.In the standard form of semidefinite programming,the decision variable is usually a semidefinite matrix.Many matrix optimization problems can be expressed by semidefinite programming,which shows that semidefinite programming is closely related to matrix optimization.In this paper,we mainly study the dual theory of convex linear semidefinite programming and the matrix optimization problem related to Schur complement.In Chapter 1,the dual theory of semidefinite programming and its application are introduced.In Chapter 2,we first study the new proof of strong duality theorem for cone optimization problems.The first proof uses the separation theorem of convex sets,the selection theorem of linear inequalities and weak cone duality theorem.The second proof uses the strong duality theorem of convex optimization and Fenchel duality.Then,the new form of expression of cone programming and its dual form are studied from the view of cone optimization and orthogonal complement space.And the form is extended to semidefinite programming.At the same time,the optimality condition is given,which is also proved in this paper.It is worth noting that the convex theory is not directly used in the proof of this optimality condition.In Chapter 3,we study the application of semidefinite programming to the matrix theory related to Schur complement.The optimality conditions of the problem are studied,and the relevant proof is given by using the strong duality theorem of semidefinite programming.Then we use the optimality condition to study the monotone property of Schur complement and the new proof of Schur complement lemma.